ALL algebra, as was pointed out by von Helmholtz nearly fifty years ago, is based upon the three following very simple propositions:
Things equal to the same thing are equal to each other.
If equals be added to equals the wholes are equal.
If unequals be added to equals the wholes are unequal.
Geometry, he adds, is founded upon a few equally obvious and simple axioms.
The science of physics, similarly, has for its foundation three fundamental conceptions: those of mass, distance and time, in terms of which all physical quantities may be expressed.
Physics, in so far as it is an exact science, deals with the relations of these so-called physical quantities; and this is true not merely of those portions of the science which are usually included under the head of physics, but also of that broader realm which consists of the entire group of the physical sciences, viz., astronomy, the physics of the heavens; chemistry, the physics of the atom; geology, the physics of the earth's crust; biology, the physics of the matter imbued with life; physics proper (mechanics, heat, electricity, sound and light).
The manner in which the three fundamental quantities L, M and T (length, mass and time) enter, in the case of a physical quantity, is given by its dimensional formula.
Thus the dimensional formula for an acceleration is LT-2 which expresses the fact that an acceleration is a velocity (a length divided by a time) divided by a time. Energy has for its dimensional formula L2MT-2; it is a force, LT-2M (an acceleration multiplied by a mass), multiplied by a distance.
Not all physical quantities, in the present state of our knowledge, can be assigned a definite dimensional formula, and this indicates that not all of physics has as yet been reduced to a clearly established mechanical basis. The dimensional formula thus affords a valuable criterion of the extent and boundaries of our strictly definite knowledge of physics. Within these boundaries we are on safe and easy ground and are dealing, independent of all speculation, with the relations between precisely defined quantities. These relations are
- An address at the International Congress of Arts and Science, St. Louis, September, 1004.
- Von Helmholtz, Populäre Wissenschaftliche Vorträge, p. 136.